Minimization with constraint on all parameters in R

r

Solution

The basic math is as follows: we start with

mu = a0 + a1*x1 + a2*x2 + a3*x3 + a4*x4

and we want to find `a0`-`a4` to minimize the SSQ between `mu` and our response variable `y`.

if we replace the last parameter (say `a4`) with (say) `C-a1-a2-a3` to honour the constraint, we end up with a new set of linear equations

mu = a0 + a1*x1 + a2*x2 + a3*x3 + (C-a1-a2-a3)*x4
   = a0 + a1*(x1-x4) + a2*(x2-x4) + a3*(x3-x4) + C*x4

(note that `a4` has disappeared ...)

Something like this (untested!) implements it in R.

Original data frame:

d <- data.frame(y=runif(20),
                x1=runif(20),
                x2=runif(20),
                x3=runif(20),
                x4=runif(20))

Create a transformed version where all but the last column have the last column "swept out", e.g. `x1 -> x1-x4; x2 -> x2-x4; ...`

dtrans <- data.frame(y=d$y,
                     sweep(d[,2:4],
                           1,
                           d[,5],
                           "-"),
                     x4=d$x4)

Rename to `tx1`, `tx2`, ... to minimize confusion:

names(dtrans)[2:4] <- paste("t",names(dtrans[2:4]),sep="")

Sum-of-coefficients constraint:

constr <- 5

Now fit the model with an offset:

lm(y~tx1+tx2+tx3,offset=constr*x4,data=dtrans)

It wouldn't be too hard to make this more general.

This requires a little more thought and manipulation than simply specifying a constraint to a canned optimization program. On the other hand, (1) it could easily be wrapped in a convenience function; (2) it's much more efficient than calling a general-purpose optimizer, since the problem is still linear (and in fact one dimension smaller than the one you started with). It could even be done with big data (e.g. `biglm`). (Actually, it occurs to me that if this is a linear model, you don't even need the offset, although using the offset means you don't have to compute `a0=intercept-C*x4` after you finish.)

Problem

I want to minimize a simple linear function `Y = x1 + x2 + x3 + x4 + x5` using ordinary least squares with the constraint that the sum of all coefficients have to equal 5. How can I accomplish this in R? All of the packages I've seen seem to allow for constraints on individual coefficients, but I can't figure out how to set a single constraint affecting coefficients. I'm not tied to OLS; if this requires an iterative approach, that's fine as well.

Original source